Semilattices of groups and nonstable K-theory of extended Cuntz limits

Abstract

We give an elementary characterization of those (abelian) semigroups $M$ that are direct limits of countable sequences of finite direct products of monoids of the form $C\cup \{0\}$ for monogenic groups $C$. This characterization involves the Riesz refinement property together with lattice-theoretical properties of the collection of subgroups of $M$, and it makes it possible to express $M$ as a certain submonoid of a direct product $S\times G$, where $S$ is a distributive semilattice with zero and $G$ is an abelian group. When applied to the monoids $V(A)$ appearing in the nonstable K-theory of C*-algebras, our results yield a full description of $V(A)$ for C*-inductive limits $A$ of finite products of full matrix algebras over either Cuntz algebras $O\_n$, where $2\le n<\infty$, or corners of $O\_\infty$ by projections, thus extending to the case including $O\_\infty$ earlier work by the authors together with K.R. Goodearl.

K-Theory, to appear

References in corpus (1)

• $K_0$ of purely infinite simple regular rings

Cited by in corpus (4)

• Representing finitely generated refinement monoids as graph monoids
• Tame and wild refinement monoids
• Relative projectivity and transferability for partial lattices
• Primely generated refinement monoids